Ukrainian Mathematical Journal, Vol. 47, No. 10, 1995

SIMULTANEOUS SIMILARITY OF CONVOLUTION OPERATORS TO POWERS OF THE OPERATOR OF INTEGRATION

R.A. ZuidwUk UDC 517.2

A necessary and sufficient condition is derived for a pair of convolution integral operators to be simultaneously similar to powers of the operator of integration. Also, simultaneous similarity of a pair of general integral operators to a pair of integral operators of convolution type is discussed. The results are compared with the matrix case.

This paper considers pairs of integral operators that are simultaneously similar to a pair of integral operators of a special form. Let A, B, C, and D be bounded linear operators on a Banach space X. The pair A, B is called

simultaneously similar to the pair C, D, if there exists a boundedly invertible operator S such that S-1AS = C

and S-1BS=D. The main theorem of this paper (Theorem 3) shows that a pair of convolution integral operators

x x

A f ( x ) = ~ a ( x - t ) f ( t ) d t , B f (x) = I b ( x - t ) f ( t ) d t o o

with certain conditions on the kernel functions a and b is simultaneously similar to a pair of powers ja, j~ of the operator of integration

X

J f (x ) = I f ( t ) d t o

if and only if A and B satisfy a simple algebraic condition. The integral operators considered here act on Lp[0, 1 ]

with 1 < p < ~o fixed. A result from [1] in this direction for pairs of finite matrices has a different outcome; see

Theorem 2. In Section 3, an abstract result (Proposition 1) is used to study simultaneous similarity of a pair of integral

operators

X X

A f ( x ) = ~ a(x, t) f( t)dt , B f (x) = ~ b(x, t ) f ( t )d t o o

to a pair of convolution integral operators (see Example 1). The proposition is also applied to pairs of lower triangular matrices (see Example 2).

1. Preliminaries

Consider the commutative algebra C[0, 1 ] of continuous functions on the standard unit interval with supremum norm

Technical University, Delft, Netherlands. Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1432-1437, October, 1995. Original article submitted March 6, 1995.

1 6 4 0 0041-5995/95/4710-1640 $12.50 �9 1996 Plenum Publishing Corporation

SIMULTANEOUS SIMILARITY OF CONVOLUTION OPERATORS TO POWERS OF THE OPERATOR OF INTEGRATION 1641

Ila[l = s u p la(x)l, 0~x~l

and convolution product

(a * b ) (x ) =

Z

f a ( x - t )b ( t )d t . o

We denote the mth convolution power by [a] m =a * ... * a. The algebra C[0, 1 ] consists of quasinilpotent

elements only, that is, for each a s C[0, 1 ], the spectral radius

lim [a]n ~n = 0. /2--.+oo

In this paper, continuous functions with the following property are of special interest: A continuous function

k e C[0, 1 ] is called of positive type k (k a positive integer) if it has absolutely continuous derivatives up to

order k, k(0) . . . . . k(k-2)(0) = 0, and k(k-1)(0) = 1. We will study integral operators with kernel functions in

C[0, 1]. If a e C[0, 1], then

A f ( x ) = ( a * f ) ( x ) = X

a ( x - t ) f ( t ) d t

o

defines a bounded operator on Lp[O, 1 ]. Indeed, II Af l lp < I1 a Ilflle for I _< p < ~. From now on, 1 < p <

will be fixed. If A and B are convolution integral operators with continuous kernel functions a and b re-

spectively, then A B is a convolution integral operator with continuous kernel function a * b. With the operator norm and this product, the convolution integral operators form a commutative Banach algebra without unit.

An example of an integral operator with convolution kernel of positive type k is the kth power of the operator of integration

J f ( x ) = (1 * f ) ( x ) = X

f f ( t ) d t

o

given by

i (X -- t ) k - 1 J k f ( x ) = ( [ 1 ] ~ * f ) ( x ) = -~--i~.v f ( t ) d t -

o

As a matter of fact, all convolution integral operators with kernel of positive type k are similar to jk by virtue of the following theorem (Theorem 5 in [2] or Theorem 7 in [3]):

Theorem 1. Let k e C[0, 1] be a continuous function of positive type k. Then the operator

K f ( x ) = x

f k ( x - t ) f ( t ) d t o

1642 R . A . . ZUIDWIJK

is similarto j k on Lp[O, 1], where 1 <p < ~ isfixed.

The situation described above has an analog for finite matrices. Indeed, the lower triangular nilpotent m x m

Jordan block

J =

rO 0 . . . . . . 0

1 0 "'.

0 1 "'. "'. 0

: "'. "'. 0 0

0 ... 0 1 0

is a matrix analog of the operator of integration. An m x m matrix of the form A = akJk+ ... + am_ 1 jm-1 with

a k r 0 is called of positive type k (assume that 1 < k < m - 1 ). Note that A is a nilpotent lower triangular Toeplitz

matrix. It is not difficult to see that A and jk are similar. Indeed, A and jk have the same Jordan canonical

form.

A more challenging problem is the following: If A and B are m x m matrices of positive types a and 13

respectively, does there exist an invertible matrix S such that S- IAS = Ja and S - I BS = J~? The following

theorem provides a sufficient condition for a pair of matrices to have this property. In [1], this theorem is stated in a

somewhat more general form; the matrices under consideration are not required to be Toeplitz. Here, we state the

version for Toeplitz matrices. Denote the greatest common divisor of two positive integers oc, 1~, by gcd ((~, 13).

Theorem 2. Let A and B be nilpotent lower triangular Toeplitz m x m matrices of positive types o~

and 13, respectively, with 1 < ct, ~ < m - 1. Write

m-1 m-1 Z = Z a k Jk' B = Z b k Jk'

k=a k=13

with aa, bB ~ O. Let gcd(c~, 13 ) = 5. I f o~ + ~ > m + 5, then there exists an invertible m x m matrix S such

that S- 1A S = j a and S- 1 B S = J ~.

Let A and B be as in Theorem 2. Obviously,

A ~/5 = s j C t ~ / 5 S - I = B a/8.

This necessary condition appears to be void. Note that o~ + [~ >_ m + 8 implies

0 ~ = ((~+ ~-13)13 > ( m + 8 - 1 3 ) ~ = ( m - 1 3 ) ( ~ - 8 ) + m8 > mS,

as [3 < m and 8 < [3, so ja~/~ = Qm, the m • m zero matrix.

2. Powers of the Operator of Integration

In this section, we state and prove the main theorem of this paper. The theorem characterizes those pairs of

convolution integral operators that are simultaneously similar to powers of the operator of integration. This result

should be compared with the matrix result of Theorem 2.

SIMULTANEOUS SIMILARITY OF CONVOLUTION OPERATORS TO POWERS OF TI'[E OPERATOR OF INTEGRATION

Theorem 3. Let a , b ~ C[0, 1]

convolution integral operators

1643

be functions of positive type a, [J, respectively, and define the

X X

Af (x ) = f a ( x - t ) f ( t ) d t , B f (x) = f b ( x - t ) f ( t ) d t , 0 o

acting on Lp [ O, 1 ], with 1 <_ p < ~ is fixed. Let 5 = gcd (or, ~3 ). The following statements are equivalent:

(i) there exists an invertible operator S on Lp[O, 1 ] such that

S-1AS = ja, S-1BS = j~;

(ii) A~/5=B r

Proof It is obvious that the first statement of the theorem implies the second one. We now prove the con- verse, using a number of claims. The first claim is taken from [2], Theorem 3 (see also Theorem 6 in [3]).

Claim 1. If K is a convolution integral operator with kernel of positive type k, then there exists a con-

volution integral operator H with kernel of positive type one such that K = H k. As in the theorem, we write

x x

-J a ( x - t ) f ( t ) d t , Bf (x) = f b ( x - t ) f ( t ) d t , A f (x ) o o

where a and b are continuous functions of positive types a and [3, respectively. Then, by Claim l, there exist

continuous functions h A and h B of positive type one such that a = [hA] a and b = [hs] ~. In particular, if one assumes that the second statement of the theorem is true, then

[hA] c~/~= [a] ~/~= [b] ~/5= [hB] a~/5. (1)

The proofs of the following two claims are rather elementary and are left to the reader.

Claim2. Let h e C 1[0, 1] be a continuously differentiable function. Then [ h ] n e Cn[0, 1] is n times

continuously differentiable and if 1 < m < n, then

k = l

Claim 3. Let q47 be a Banach algebra consisting of quasinilpotent elements only. Let v, w ~ q4 ? be such that

1) k _= W k.

k = l = k

Then v = w.

1644

Claim 3 implies, with q / i ]=C[0 ,1 ] , v = h ~ and w = h 8, and r e = n ,

hB(0 ) = 1 implies that h A = h B. Write h = h A = h B. Apply Theorem 1 to

that ha hl

R. A. ZUIDWLIK

Then h A (0) =

H f ( x ) =

X

I h ( x - t ) f ( t ) d t

o

to obtain a similarity S such that S- I HS = J. Then, accordingly,

S - 1 A S = S - 1 H a S = Ja, S - 1 B S = S - 1 H ~ S = J~.

The theorem is proved.

The situation for lower triangular Toeplitz m x m matrices appears to be quite different from the situation

described in Theorem 3. Indeed, the lower triangular Toeplitz 7 x 7 matrices

A =

0 0 0 0 0 0 0 ~

0 0 0 0 0 0 0

1 0 0 0 0 0 0

1 1 0 0 0 0 0

1 1 1 0 0 0 0

1 1 1 1 0 0 0

1 1 1 1 1 0 0

, B =

/ 0 0 0 0 0 0 0 ~

0 0 0 0 0 0 0

0 0 0 0 0 0 0

1 0 0 0 0 0 0

1 1 0 0 0 0 0

1 1 1 0 0 0 0

1 1 1 1 0 0 0

satisfy A 3 = B 2, but there exists no invertible 7 x 7 matrix S such that both S- 1A S and S - 1 B S are powers of

the nilpotent lower triangular 7 x 7 Jordan block. Details are left to the reader.

3. Convolut ion Integral Operators

Another simultaneous similarity problem is the following: Let G = { (x, y) I 0 <_ y < x < 1 } denote the triangle

in the real plane, and let a, b ~ C(G) be continuous functions on G. The integral operators

x X

A f ( x ) = f a(x, t ) f ( t ) d t , B f ( x ) = f b(x, t ) f ( t ) d t

o o

act as bounded operators on the space Lp[0, 1 ] with 1 < p < o~ fixed. The question is whether there exists an

invertible operator S on Lp[0, 1 ] such that both S - 1 A S and S - 1 B S are convolution integral operators. The

following rather abstract proposition based on material in [4] provides sufficient conditions for simultaneous similarity to a pair of convolution integral operators.

Propos i t ion 1. Let A be an algebra, let B C_ A 1 C_ A be subsets, and let ~ : A 1 --~ A be a derivation.

I f there exists an M ~ .f~ such that

(i) ~B = M B - B M for all B ~ B;

(ii) ~ S = M S for an invertible S e A 1,

S IMULTANEOUS SIMILARITY OF CONVOLUTION OPERATORS TO POWERS OF THE OPERATOR OF INTEGRATION 1645

then S -1 BS CKer(O).

Proof. Let F = S- 1 BS for some B e B and invertible S ~ A 1 as in the proposition. We need to prove that

OF= O, the zero operator. Write S F = B S and apply ~ on both sides to obtain

(OS)F + SOF = (OB)S + BOS.

Since OB = MB - B M and OS = MS, we get

M S F + SOF = MB S - BMS + BMS,

SO

The proposition is proved.

OF = S - 1 M ( B S - S F ) = O.

We will describe two situations in which Proposition 1 is of some use.

Example 1. Let C(G) denote the continuous functions on G and let C 1 (G) C C(G) be the functions with

continuous partial derivatives on G. Supplied with the supremum norm and with product

(ab ) ( x , y ) =

C(G) becomes a Banach algebra. The derivation

has the kernel

x

I a(x, t )b(t , y)dt,

Y

0 0 a = + : c (c) c ( v )

ay

Ker(O) = { a ( x , y ) e C l ( G ) l a ( x , y ) = g t ( x -y ) forsome f i e c l [ 0 , l ] } .

Fix 1 < p < ~o. Note that each a ~ C(G) corresponds to an integral operator

A f ( x ) =

which acts o n Lp[O, 1 ]. Define the unital algebra

and the subset

X

I a(x, t ) f ( t )d t ,

0

A = { L I + A ] L ~ (E, a ~ C(G)}

A 1 = { ~ I + A [ ~ , ~ ~, a ~ CI(G)}.

Let A o and B o be elements in A 1, and define the set B= {A o, Bo}. Further, define the derivation 0" A 1 ---> A

1646 R.A. ZUIDWLIK

by ~()~I+ A) = OA/Ox + OA/Oy. If the two "derivational equations" in Proposition 1 are satisfied, then S-1Ao S

and S- IBo S are in Ker(O) and, hence, are convolution integral operators.

Example 2. Consider the algebra A_ of lower triangular m x m matrices. Let J ~ A_ denote the nilpotent

lower triangular m x m Jordan block. Define the derivation 2" A ~ A by ~ A = J A - A J. It follows that

Ker(~) is the algebra of lower triangular Toeplitz matrices. Take A = A 1 = A_ and let A 0 and B 0 be lower

triangular m x m matrices. Define the set B = {A 0' B0}.

If the two "derivational equations" in Proposition 1 are satistied, then S-1 A 0S and S- 1 B 0S are in Ker (~) and, hence, are lower triangular Toeplitz matrices.

The "derivational equations" in Proposition 1 seem to be of independent interest. A further understanding of these equations and possible applications to the notion of simultaneous similarity is an aim of future research.

The author would like to thank H. Bart and M. M. Malamud for their helpful comments and suggestions.

REFERENCES

t. H. Bart and G. Ph. A. Thijsse, "Complementary triangular forms of upper triangular Toeplitz matrices," Operator Theory: Adv. Appl., 40, 133-149 (1989).

2. M.M. Malamud, "Sufficient conditions for the linear equivalence of Volterra operators," Teor. Funkts., Funkts. Anal. Prilozhen., Issue 23, 59--69 (1975).

3. M.M. Malamud and E. R. Tsekanovskii, "Criteria for linear equivalence of Volterra operators in the scale Lp [0, 1 ] ( 1 _< p < ~),"

Izv. Akad. Nauk SSSR, Ser. Mat., 41, No. 4, 725-748 (1977). 4. M.M. Malamud and E. R. Tsekanovskii, "On the linear equivalence of Volterra operators in Banach spaces," Transl. Amer. Math.

Soc., 115, 187-197 (1980).